In a large bowl, stir together oats, water and butter - cover and set aside for 10 minutes. Add cornmeal, sugar, bullion, milk,cheese and egg - still well to combine. Add all-purpose flour and mix until combined. Add whole-wheat flour, stirring to form a stiff dough.

Scoop the dough onto a floured surface and knead, adding additional whole-wheat flour, until the dough is smooth and no longer sticky. Roll out the dough to a thickness of about 1/4". Cut out as many pieces as you can with your cutters and place onto parchment-lined baking sheet. Pull the scraps together, re-roll dough and continue cutting out shapes until the dough has been used up.

Bake, rotating sheets halfway through, until browned, about 30 to 45 minutes. Remove from the oven and transfer the dog treats to a wire rack to cool completely.